Abstract
The purpose of this paper is to introduce the concepts of total asymptotically strictly pseudocontractive semigroup, asymptotically strictly pseudocontractive semigroup etc., and to prove some strong convergence theorems of the explicit iteration process for these kinds of semigroups in arbitrary Banach spaces. The results presented in the paper extend and improve some recent results announced in the current literature.
Highlights
Introduction and preliminaries LetE be a real Banach space, E* be the dual space of E
Let T : C → C be a mapping, we denote by F(T) the set of fixed points of T
A one-parameter family := {T(t) : t ≥ } of self-mappings of C is said to be a nonexpansive semigroup if the following conditions are satisfied: (i) T(t + t )x = T(t )T(t )x, for any t, t ∈ + and x ∈ C; (ii) T( )x = x for each x ∈ C; (iii) for each x ∈ C, t → T(t)x is continuous; (iv) for any t ≥, T(t) is a nonexpansive mapping on C, that is, for any x, y ∈ C, T(t)x – T(t)y ≤ x – y
Summary
Introduction and preliminaries LetE be a real Banach space, E* be the dual space of E. If the family := {T(t) : t ≥ } satisfies conditions (i)-(iii), it is: (a) Pseudocontractive semigroup if for any x, y ∈ C, there exists j(x – y) ∈ J(x – y) such that
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